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3 A GAUSS-KUZMIN THEOREM AND RELATED

QUESTIONS FOR θ-EXPANSIONS

Gabriela Ileana Sebe∗

Politehnica University of Bucharest, Faculty of Applied Sciences,

Splaiul Independentei 313, 060042, Bucharest, Romania

and

Dan Lascu†

Mircea cel Batran Naval Academy, 1 Fulgerului, 900218 Constanta, Romania

May 19, 2014

Abstract

Using the natural extension for θ-expansions, we give an infinite-order-chain representation of the sequence of the incomplete quotientsof these expansions. Together with the ergodic behavior of a certainhomogeneous random system with complete connections, this allowsus to solve a variant of Gauss-Kuzmin problem for the above fractionexpansion.

Mathematics Subject Classifications (2010).Key words: continued fractions, θ-expansions, Perron-Frobenius operator,random system with complete connections, Gauss-Kuzmin problem.

1 Introduction

During the last fifty years a large amount of research has been devoted tothe study of various algorithms for the representation of real numbers bymeans of sequences of integers. Motivated by problems in random contin-ued fraction expansions (see [1]), Chakraborty and Rao [4] have initiated asystematic study of the continued fraction expansion of a number in terms

∗e-mail: igsebe@yahoo.com.†e-mail: lascudan@gmail.com.

of an irrational θ ∈ (0, 1). This new expansion of positive reals, differentfrom the regular continued fraction expansion is called θ-expansion.

The purpose of this paper is to solve a Gauss-Kuzmin problem for θ-expansions. In order to solve the problem, we apply the theory of randomsystems with complete connections extensively studied by Iosifescu and Grig-orescu [7]. First we outline the historical framework of this problem. InSection 1.2, we present the current framework. In Section 1.3, we reviewknown results.

1.1 Gauss’ Problem

One of the first and still one of the most important results in the metricaltheory of continued fractions is the so-called Gauss-Kuzmin theorem. Anyirrational 0 < x < 1 can be written as the infinite regular continued fraction

x =1

a1 +1

a2 +1

a3 +.. .

:= [a1, a2, a3, . . .], (1.1)

where an ∈ N+ := {1, 2, 3, . . .} [8]. Such integers a1, a2, . . . are called incom-plete quotients (or continued fraction digits) of x. The metrical theory ofcontinued fraction expansions started on 25th October 1800, with a note byGauss in his mathematical diary [3]. Define the regular continued fraction(or Gauss) transformation τ on the unit interval I := [0, 1] by

τ(x) =

1

x−

⌊

1

x

⌋

if x 6= 0,

0 if x = 0,

(1.2)

where ⌊·⌋ denotes the floor (or entire) function. With respect to the asymp-totic behavior of iterations τn = τ ◦ · · · ◦ τ (n-times) of τ , Gauss wrote (inmodern notation) that

limn→∞

λ (τn ≤ x) =log(1 + x)

log 2, x ∈ I, (1.3)

where λ denotes the Lebesgue measure on I. In 1812, Gauss asked Laplace[3] to estimate the n-th error term en(x) defined by

en(x) := λ(τ−n[0, x]) − log(1 + x)

log 2, n ≥ 1, x ∈ I. (1.4)

2

This has been called Gauss’ Problem. It received first solution more than acentury later, when R.O. Kuzmin [10] showed in 1928 that en(x) = O(q

√n)

as n → ∞, uniformly in x with some (unspecified) 0 < q < 1. This has beencalled the Gauss-Kuzmin theorem or the Kuzmin theorem.

One year later, using a different method, Paul Levy [12] improvedKuzmin’s result by showing that |en(x)| ≤ qn for n ∈ N+, x ∈ I, withq = 3.5− 2

√2 = 0.67157.... For such historical reasons, the Gauss-Kuzmin-

Levy theorem is regarded as the first basic result in the rich metrical theoryof continued fractions. An advantage of the Gauss-Kuzmin-Levy theoremrelative to the Gauss-Kuzmin theorem is the determination of the value ofq.

To this day the Gauss transformation, on which metrical theory of reg-ular continued fraction is based, has fascinated researchers from variousbranches of mathematics and science with many applications in computerscience, cosmology and chaos theory [5]. In the last century, mathemati-cians broke new ground in this area. Apart from the regular continued frac-tion expansion, very many other continued fraction expansions were studied[13, 15].

By such a development, generalizations of these problems for non-regularcontinued fractions are also called as the Gauss-Kuzmin problem and theGauss-Kuzmin-Levy problem [9, 11, 16, 17, 18].

1.2 θ-expansions

For a fixed θ ∈ (0, 1), we start with a brief review of continued fractionexpansion with respect to θ, analogous to the regular continued fractionexpansion which corresponds to the case θ = 1.

For x ∈ (0,∞) let

a0 := max{n ≥ 0 : nθ ≤ x}.

If x equals a0θ, we writex := [a0θ].

If not, define r1 by

x := a0θ +1

r1

where 0 < 1/r1 < θ. Then r1 > 1/θ ≥ θ and let

a1 := max{n ≥ 0 : nθ ≤ r1}.

3

If r1 = a1θ, then we write

x := [a0θ, a1θ],

i.e.,

x = a0θ +1

a1θ.

If a1θ < r1, define r2 by

r1 := a1θ +1

r2

where 0 < 1/r2 < θ. So, r2 > 1/θ ≥ θ and let

a2 := max{n ≥ 0 : nθ ≤ r2}.

In this way, either the process terminates after a finite number of steps orit continues indefinitely. Following standard notation, in the first case wewrite

x := [a0θ; a1θ, . . . , anθ] (1.5)

and we call this the finite continued fraction expansion of x with respect toθ (terminating at the n-stage). In the second case, we write

x := [a0θ; a1θ, a2θ, . . .] (1.6)

and we call this the infinite continued fraction expansion of x with respectto θ.

When 0 < x < θ, we have a0 = 0 and instead of writing

x := [0; a1θ, a2θ, . . .], (1.7)

we simply writex := [a1θ, a2θ, . . .] (1.8)

which is the same in the usual notation

x =1

a1θ +1

a2θ +1

a3θ +.. .

. (1.9)

Such an’s are also called incomplete quotients (or continued fraction digits)of x with respect to the expansion in (1.9).

4

In general, the θ-expansion of a number x > 0 is

a0θ + [0; a1θ, a2θ, . . .] := [a0θ; a1θ, a2θ, . . .] (1.10)

where a0 = ⌊x/θ⌋.For x ∈ [0, θ], the θ-continued fraction expansion of x in (1.9) leads to

an analogous transformation of Gauss map τ in (1.2). A natural question iswhether this new transformation admits an absolutely continuous invariantprobability like the Gauss measure in the case θ = 1. Until now, the invariantmeasure was identified only in the particular case θ2 = 1/m, m a positiveinteger [4].

Motivated by this argument, since the invariant measure is a crucial toolin our approach, in the sequel we will consider only the case θ2 = 1/mwith m a positive integer. Then [a1θ, a2θ, a3θ, . . .] is the θ-expansion of anyx ∈ [0, θ] if and only if the following conditions hold:

(i) an ≥ m for any m ∈ N+;

(ii) in case when x has a finite expansion, i.e., x = [a1θ, a2θ, a3θ, . . . , anθ],then an ≥ m+ 1.

This continued fraction is treated as the following dynamical system.

Definition 1.1. Let θ ∈ (0, 1) and m ∈ N+ such that θ2 = 1/m.

(i) The measure-theoretical dynamical system ([0, θ],B[0,θ], Tθ) is definedas follows: B[0,θ] denotes the σ-algebra of all Borel subsets of [0, θ],and Tθ is the transformation

Tθ : [0, θ] → [0, θ]; Tθ(x) :=

1

x− θ

⌊

1

xθ

⌋

if x ∈ (0, θ],

0 if x = 0.

(1.11)

(ii) In addition to (i), we write ([0, θ],B[0,θ], γθ, Tθ) as ([0, θ],B[0,θ], Tθ) withthe following probability measure γθ on ([0, θ],B[0,θ]):

γθ(A) :=1

log (1 + θ2)

∫

A

θdx

1 + θx, A ∈ B[0,θ]. (1.12)

By using Tθ, the sequence (an)n∈N+in (1.9) is obtained as follows:

an = an(x) = a1(

T n−1θ (x)

)

, n ∈ N+, (1.13)

5

with T 0θ (x) = x and

a1 = a1(x) =

{

⌊ 1xθ⌋ if x 6= 0,

∞ if x = 0.(1.14)

In this way, Tθ gives the algorithm of θ-expansion.

Proposition 1.2. Let ([0, θ],B[0,θ], γθ, Tθ) be as in Definition 1.1(ii).

(i) ([0, θ],B[0,θ], γθ, Tθ) is ergodic.

(ii) The measure γθ is invariant under Tθ, that is, γθ(A) = γθ(T−1θ (A))

for any A ∈ B[0,θ].

Proof. See Section 8 in [4]. �

By Proposition 1.2(ii), ([0, θ],B[0,θ], γθ, Tθ) is a “dynamical system” inthe sense of Definition 3.1.3 in [2].

1.3 Known results and applications

In this subsection we recall known results and their applications for θ-expansions.

1.3.1 Known results

Let 0 < θ < 1. Define the n-th order convergent [a1θ, a2θ, . . . , anθ] of x ∈[0, θ] by truncating the θ-expansion in (1.9). Thus, Chakraborty and Raoproved in [4] that

[a1θ, a2θ, . . . , anθ] → x, n → ∞. (1.15)

In what follows the stated identities hold for all n in case x has an infiniteθ-expansion and they hold for n ≤ k in case x has a finite θ-expansionterminating at the k-th stage.

To this end, define real functions pn(x) and qn(x), for n ∈ N+, by

pn(x) := an(x)θpn−1(x) + pn−2(x), (1.16)

qn(x) := an(x)θqn−1(x) + qn−2(x), (1.17)

with p−1(x) := 1, p0(x) := 0, q−1(x) := 0 and q0(x) := 1. By induction, wehave

pn−1(x)qn(x)− pn(x)qn−1(x) = (−1)n, n ∈ N := N+ ∪ {0}. (1.18)

6

By using (1.16) and (1.17), we can verify that

x =pn(x) + T n

θ (x)pn−1(x)

qn(x) + T nθ (x)qn−1(x)

, n ≥ 1. (1.19)

By taking T nθ (x) = 0 in (1.19), we obtain [a1θ, a2θ, . . . , anθ] = pn(x)/qn(x).

Using (1.18) and (1.19) we obtain

x− pn(x)

qn(x)=

(−1)n+1T nθ (x)

qn(x)(qn(x) + T nθ (x)qn−1(x))

, n ≥ 1. (1.20)

By applying 0 ≤ T nθ ≤ θ to (1.20), we can verify that

1

qn(x)(qn+1(x) + θqn(x))≤

∣

∣

∣

∣

x− pn(x)

qn(x)

∣

∣

∣

∣

≤ 1

qn(x)qn+1(x). (1.21)

From (1.17), we have that qn(x) ≥ θ, n ∈ N+. Further, also from (1.17) andby induction we have that

qn(x) ≥⌊n

2

⌋

θ2. (1.22)

Finally, (1.15) follows from (1.21) and (1.22).

1.3.2 Application to ergodic theory

Similarly to classical results on regular continued fractions, using the er-godicity of Tθ and Birkhoff’s ergodic theorem [6], a number of results wereobtained.

For qn in (1.17), its asymptotic growth rate β is defined as

β = limn→∞

1

nlog qn. (1.23)

This is a Levy result and Chakraborty and Rao [4] obtained that β is a finitenumber

β =1

1 + θ2

∫ θ

0

θ log x

1 + xθdx. (1.24)

They also give a Khintchin result, i.e., the asymptotic value of the arithmeticmean of a1, a2, . . . , an where a1 and an are given in (1.14) and (1.13). Wehave

limn→∞

a1 + a2 + . . .+ ann

= ∞. (1.25)

7

It should be stressed that the ergodic theorem does not yield any in-formation on the convergence rate in the Gauss problem that amounts tothe asymptotic behavior of µ(T−n

θ ) as n → ∞, where µ is an arbitraryprobability measure on B[0,θ].

It is only very recently that there has been any investigation of the met-rical properties of the θ-expansions. Thus, the results obtained in this paperallow to a solution of a Gauss-Kuzmin type problem. We may emphasizethat, to our knowledge, Theorem 6.1 is the first Gauss-Kuzmin result provedfor θ-expansions. Our solution presented here is based on the ergodic be-havior of a certain random system with complete connections.

The paper is organized as follows. In Section 2, we show the probabilitystructure of (an)n∈N+

under the Lebesgue measure by using the Broden-Borel-Levy formula. In Section 3, we consider the so-called natural ex-tension of ([0, θ],B[0,θ], γθ, T[0,θ]) [14]. In Section 4, we derive its Perron-Frobenius operator under different probability measures on ([0, θ],B[0,θ]).Especially, we derive the asymptotic behavior for the Perron-Frobenius op-erator of (([0, θ],B[0,θ], γθ, T[0,θ]). In Section 5, we study the ergodicity ofthe associated random system with complete connections (RSCC for short).In Section 6, we solve a variant of Gauss-Kuzmin problem for θ-expansions.By using the ergodic behavior of the RSCC introduced in Section 5, wedetermine the limit of the sequence (µ(T n

θ < x) )n≥1 of distributions asn → ∞.

2 Prerequisites

Roughly speaking, the metrical theory of continued fraction expansions isabout the sequence (an)n∈N+

of incomplete quotients and related sequences[8]. As remarked earlier in the introduction we will adopt a similar strategyto that used for regular continued fractions. We begin with a Broden-Borel-Levy formula for θ-expansions. Then some consequences of it to be used inthe sequel are also derived.

In this section let us fix 0 < θ < 1, θ2 = 1/m, m ∈ N+.For x ∈ [0, θ] consider an = an(x), n ∈ N+, as in (1.13) and (1.14).

Putting Nm := {m,m + 1, . . .}, m ∈ N+, the incomplete quotients an, n ∈N+, take positive integer values in Nm.

For any n ∈ N+ and i(n) = (i1, . . . , in) ∈ Nnm, define the fundamental

interval associated with i(n) by

I(i(n)) = {x ∈ [0, θ] : ak(x) = ik for k = 1, . . . , n}, (2.1)

8

where I(i(0)) = [0, θ]. For example, for any i ∈ Nm we have

I (i) = {x ∈ [0, θ] : a1(x) = i} =

(

1

(i+ 1)θ,1

iθ

)

. (2.2)

We will write I(a1, . . . , an) = I(

a(n))

, n ∈ N+. If n ≥ 1 and in ∈ Nm,then we have

I(a1, . . . , an) = I(

i(n))

. (2.3)

From the definition of Tθ and (1.19), we have

I(a(n)) = (u(a(n)), v(a(n))), (2.4)

where u(a(n)) and v(a(n)) are defined as

u(

a(n))

:=

pn + θpn−1

qn + θqn−1if n is odd,

pnqn

if n is even,

(2.5)

and

v(

a(n))

:=

pnqn

if n is odd,

pn + θpn−1

qn + θqn−1if n is even,

(2.6)

where pn := pn(x) and qn := qn(x) are defined in (1.16) and (1.17), respec-tively.

Let λθ denote a Lebesgue measure on [0, θ]. Using (1.18) we get

λθ

(

I(

a(n)))

=1

θ

∣

∣

∣

∣

pnqn

− pn + θpn−1

qn + θqn−1

∣

∣

∣

∣

=1

qn(qn + θqn−1). (2.7)

To derive the so-called Broden-Borel-Levy formula [7, 8] for θ-expansions, let us define (sn)n∈N by

s0 := 0, sn := qn−1/qn, n ≥ 1. (2.8)

From (1.17), sn = 1/(anθ + sn−1) for n ≥ 1. Hence

sn =1

anθ +1

an−1θ +.. . +

1

a1θ

:= [anθ, an−1θ, . . . , a1θ], (2.9)

for n ≥ 1.

9

Proposition 2.1 (Broden-Borel-Levy-type formula). Let λθ denote theLebesgue measure on [0, θ]. For any n ∈ N+, the conditional probabilityλθ(T

nθ < x|a1, . . . , an) is given as follows:

λθ(Tnθ < x|a1, . . . , an) =

(snθ + 1)x

θ(snx+ 1), x ∈ [0, θ], (2.10)

where sn is defined in (2.8) and a1, . . . , an are as in (1.13) and (1.14).

Proof. By definition, we have

λθ(Tnθ < x|a1, . . . , an) =

λθ ((Tnθ < x) ∩ I(a1, . . . , an))

λθ (I(a1, . . . , an))(2.11)

for any n ∈ N+ and x ∈ [0, θ]. Using (1.19) and (2.4) we get

λθ ((Tnθ < x) ∩ I(a1, . . . , an)) =

1

θ

∣

∣

∣

∣

pnqn

− pn + xpn−1

qn + xqn−1

∣

∣

∣

∣

=x

qn(qn + xqn−1)θ.

From this and (2.7) it follows that

λθ (Tnθ < x|a1, . . . , an) =

λθ ((Tnθ < x) ∩ I(a1, . . . , an))

λθ (I(a1, . . . , an))

=x (qn + θqn−1)

(qn + xqn−1)θ=

x(snθ + 1)

(snx+ 1)θ, (2.12)

for any n ∈ N+ and x ∈ [0, θ]. �

The Broden-Borel-Levy formula allows us to determine the probabilitystructure of incomplete quotients (an)n∈N+

under λθ.

Proposition 2.2. For any i ∈ Nm and n ∈ N+, we have

λθ(a1 = i) =m

i(i+ 1), λθ (an+1 = i|a1, . . . , an) = Pi(sn), (2.13)

where (sn)n∈N+is defined in (2.8), and

Pi(x) :=xθ + 1

(x+ iθ)(x+ (i+ 1)θ). (2.14)

10

Proof. From (2.2), the case λθ(a1 = i) holds. For n ∈ N+ and x ∈ [0, θ],we have T n

θ (x) = [an+1θ, an+2θ, . . .]. Using (2.10) we obtain

λθ( an+1 = i | a1, . . . , an ) = λθ

(

T nθ ∈

(

1

(i+ 1)θ,1

iθ

]

| a1, . . . , an)

.

=(snθ + 1) 1

iθ

θ(sn1iθ+ 1)

−(snθ + 1) 1

(i+1)θ

θ(sn1

(i+1)θ + 1)

= Pi(sn). (2.15)

�

Remark 2.3. (i) It is easy to check that

∞∑

i=m

Pi(x) = 1 for any x ∈ [0, θ]. (2.16)

(ii) Proposition 2.2 is the starting point of an approach to the metricaltheory of θ-expansions via dependence with complete connections (see[7], Section 5.2)

Corollary 2.4. The sequence (sn)n∈N+with s0 = 0 is an [0, θ]-Markov chain

on ([0, θ],B[0,θ], λθ) with the following transition mechanism: from state s thepossible transitions are to any state 1/(s+ iθ) with corresponding transitionprobability Pi(s), i ∈ Nm.

3 An infinite-order-chain representation

In this section we introduce the natural extension Tθ of Tθ in (1.11) anddefine extended random variables according to Chap.1.3.3 of [8]. Then wegive an infinite-order-chain representation of the sequence of the incompletequotients for θ-expansions.

3.1 Natural extension

Let ([0, θ],B[0,θ], Tθ) be as in Definition 1.1(i). Let be [0, θ]2 := [0, θ]× [0, θ]and the square space ([0, θ]2,B2

[0,θ]) := ([0, θ],B[0,θ])× ([0, θ],B[0,θ]).

11

Definition 3.1. The natural extension [14] of ([0, θ],B[0,θ], Tθ) is

([0, θ]2,B2[0,θ], Tθ) where the transformation Tθ of ([0, θ]2,B2

[0,θ]) is defined by

Tθ : [0, θ]2 → [0, θ]2; Tθ(x, y) :=

(

Tθ(x),1

a1(x)θ + y

)

, (x, y) ∈ [0, θ]2.

(3.1)

This is a one-to-one transformation of [0, θ]2 with the inverse

(Tθ)−1(x, y) =

(

1

a1(y)θ + x, Tθ(y)

)

, (x, y) ∈ [0, θ]2 . (3.2)

Iterations of (3.1) and (3.2) are given as follows for each n ≥ 2:(

Tθ

)n(x, y) = (T n

θ (x), [an(x)θ, an−1(x)θ, . . . , a2(x)θ, a1(x)θ + y] ), (3.3)

(

Tθ

)−n(x, y) = ( [an(y)θ, an−1(y)θ, . . . , a2(y)θ, a1(y)θ + x], T n

θ (y) ). (3.4)

For γθ in (1.12), define its extended measure γθ on ([0, θ]2,B2[0,θ]) as

γθ(B) :=1

log(1 + θ2)

∫∫

B

dxdy

(1 + xy)2, B ∈ B2

[0,θ]. (3.5)

Thenγθ(A× [0, θ]) = γθ([0, θ]×A) = γθ(A) (3.6)

for any A ∈ B[0,θ].

Proposition 3.2. The measure γθ is preserved by Tθ.

Proof. We show that γθ((Tθ)−1(B)) = γθ(B) for any B ∈ B2

[0,θ]. Since Tθ

is invertible on [0, θ]2, the last equation is equivalent to

γθ(Tθ(B)) = γθ(B) for any B ∈ B2[0,θ]. (3.7)

Recall fundamental interval in (2.1). Since the collection of Cartesian prod-ucts I

(

i(n))

×I(

j(t))

, i(n) ∈ Nnm, j(t) ∈ N

tm, n, t ∈ N, generates the σ-algebra

B2[0,θ], it is enough to show that

γθ

(

Tθ

(

I(

i(n))

× I(

j(t))))

= γθ

(

I(

i(n))

× I(

j(t)))

(3.8)

for any i(n) ∈ Nnm, j(t) ∈ N

tm, n, t ∈ N. It follows from (3.6) and Proposition

1.2(ii) that (3.8) holds for n = 0 and t ∈ N. If n ∈ N+, then it is easy to seethat

Tθ

(

I(

i(n))

× I(

j(t)))

= I(i2, . . . , in)× I(i1, j1, . . . , jt), t ∈ N+, (3.9)

12

where I(i2, . . . , in) equals [0, θ] for n = 1. Also, if I(

i(n))

= (a, b) ⊂ [0, θ]

and I(

j(t))

= (c, d) ⊂ [0, θ], then

I(i2, . . . , in) =

(

1

b− i1θ,

1

a− i1θ

)

and

I(i1, j1, . . . , jt) =

(

1

d+ i1θ,

1

c+ i1θ

)

.

A simple computation yields

γθ ((a, b)× (c, d)) =1

log(1 + θ2)log

(ac+ 1)(bd + 1)

(ad+ 1)(bc + 1)(3.10)

and then

γθ

((

1

b− i1θ,

1

a− i1θ

)

×(

1

d+ i1θ,

1

c+ i1θ

))

=1

log(1 + θ2)log

((1/b − i1θ)/(d+ i1θ) + 1)((1/a − i1θ)/(c+ i1θ) + 1)

((1/b − i1θ)/(c+ i1θ) + 1)((1/a − i1θ)/(d+ i1θ) + 1)

=1

log(1 + θ2)log

(ac+ 1)(bd + 1)

(ad+ 1)(bc + 1), (3.11)

that is, (3.8) holds. �

3.2 Extended random variables

With respect to Tθ in (3.1), define extended incomplete quotients al(x, y),l ∈ Z := {. . . ,−2,−1, 0, 1, 2, . . .}, (x, y) ∈ [0, θ]2 by

al+1(x, y) := a1((Tθ)l(x, y) ), l ∈ Z, (3.12)

witha1(x, y) = a1(x), (x, y) ∈ [0, θ]2. (3.13)

Remark 3.3. (i) Since Tθ is invertible it follows that al(x, y) in (3.12)is also well-defined for l ≤ 0. By (3.3) and (3.4), we have

an(x, y) = an(x), a0(x, y) = a1(y), a−n(x, y) = an+1(y) (3.14)

for any n ∈ N+ and (x, y) ∈ [0, θ]2.

(ii) From Proposition 3.2, the doubly infinite sequence (al(x, y))l∈Z isstrictly stationary (i.e., its distribution is invariant under a shift ofthe indices) under γθ.

13

The following theorem will play a key role in the sequel.

Theorem 3.4. For any x ∈ [0, θ] we have

γθ([0, x] × [0, θ] | a0, a−1, . . .) =(aθ + 1)x

(ax+ 1)θγθ-a.s., (3.15)

where a := [a0θ, a−1θ, . . .] (= the θ-expansion with incomplete quotients a0,a1, . . .).

Proof. Let In denote the fundamental interval I(a0, a−1, . . . , a−n) for n ∈ N.We have

γθ([0, x] × [0, θ] | a0, a−1, . . .) = limn→∞

γθ([0, x]× [0, θ] | a0, . . . , a−n) γθ-a.s.

(3.16)and

γθ([0, x]× [0, θ] | a0, . . . , a−n) =γθ([0, x] × In)

γθ([0, θ] × In)

=1

γθ(In)

1

log(1 + θ2)

∫

In

dy

∫ x

0

du

(uy + 1)2

=1

γθ(In)

1

θ

∫

In

x(1 + yθ)

1 + xydγθ(y)

=x(ynθ + 1)

(xyn + 1)θ, (3.17)

for some yn ∈ In. Since

limn→∞

yn = [a0θ, a−1θ, . . .] = a, (3.18)

the proof is completed. �

The probability structure of (al)l∈Z under γθ is given as follows.

Corollary 3.5. For any i ∈ Nm, we have

γθ(a1 = i| a0, a−1, . . .) = Pi(a) γθ-a.s., (3.19)

where a = [a0θ, a−1θ, . . .] and the functions Pi, i ∈ Nm, are defined by (2.14).

Proof. Let In be as in the proof of Theorem 3.4. We have

γθ(a1 = i | a0, a−1, . . .) = limn→∞

γθ(a1 = i | In). (3.20)

14

We have

(a1 = i) =

(

1

(i+ 1)θ,1

iθ

]

× [0, θ], i ∈ Nm. (3.21)

Now

γθ

((

1

(i+ 1)θ,1

iθ

]

× [0, θ]

∣

∣

∣

∣

In

)

=γθ

((

1(i+1)θ ,

1iθ

]

× In

)

γθ([0, θ]× In)

=1

γθ(In)

∫

In

Pi(y) dγθ(y)

= Pi(yn), (3.22)

for some yn ∈ In. From (3.18), the proof is completed. �

Remark 3.6. The strict stationarity of (al)l∈Z, under γθ implies that

γθ(al+1 = i | al, al−1, . . .) = Pi(a) γθ-a.s. (3.23)

for any i ∈ Nm and l ∈ Z, where a = [alθ, al−1θ, . . .]. The last equationemphasizes that (al)l∈Z is an infinite-order-chain in the theory of dependencewith complete connections (see [7], Section 5.5).

Define extended random variables (sl)l∈Z as sl := [alθ, al−1θ, . . .], l ∈Z. Clearly, sl = s0 ◦ (Tθ)

l, l ∈ Z. It follows from Proposition 3.2 andCorollary 3.5 that (sl)l∈Z is a strictly stationary [0, θ]-valued Markov processon ([0, θ]2,B2

[0,θ], γθ) with the following transition mechanism: from state s ∈[0, θ] the possible transitions are to any state 1/(s+ iθ) with correspondingtransition probability Pi(s), i ∈ Nm. Clearly, for any l ∈ Z we have

γθ(sl < x) = γθ(s0 < x) = γθ([0, θ]× [0, x)) = γθ([0, x)), x ∈ [0, θ]. (3.24)

Motivated by Theorem 3.4, we shall consider the one-parameter fam-ily {γθ,a : a ∈ [0, θ]} of (conditional) probability measures on ([0, θ],B[0,θ])defined by their distribution functions

γθ,a([0, x]) :=(aθ + 1)x

(ax+ 1)θ, x, a ∈ [0, θ]. (3.25)

Note that γθ,0 = λθ.For any a ∈ [0, θ] put

s0,a := a, sn,a :=1

anθ + sn−1,a, n ∈ N+. (3.26)

15

Remark 3.7. It follows from the properties just described of the process(sl)l∈Z that the sequence (sn,a)n∈N+

is an [0, θ]-valued Markov chain on([0, θ],B[0,θ], γθ,a) which starts at s0,a := a and has the following transitionmechanism: from state s ∈ [0, θ] the possible transitions are to any state1/(s + iθ) with corresponding transition probability Pi(s), i ∈ Nm.

4 An operatorial treatment

Let ([0, θ],B[0,θ], γθ, Tθ) be as in Definition 1.1(ii). The Gauss-Kuzmin prob-lem for the transformation Tθ can be approached in terms of the associatedPerron-Frobenius operator.

Let µ be a probability measure on ([0, θ],B[0,θ]) such that µ((Tθ)−1(A)) =

0 whenever µ(A) = 0 for any A ∈ B[0,θ]. For example, this condition issatisfied if Tθ is µ-preserving, that is, µ(Tθ)

−1 = µ. Let

L1µ := {f : [0, θ] → C :

∫ θ

0|f |dµ < ∞}.

The Perron-Frobenius operator of Tθ under µ is defined as the boundedlinear operator Uµ which takes the Banach space L1

µ into itself and satisfiesthe equation

∫

A

Uµf dµ =

∫

(Tθ)−1(A)f dµ for all A ∈ B[0,θ], f ∈ L1

µ. (4.1)

Proposition 4.1. (i) The Perron-Frobenius operator U := Uγθ of Tθ

under the invariant probability measure γθ is given a.e. in [0, θ] by theequation

Uf(x) =∑

i≥m

Pi(x) f(ui(x)), m ∈ N+, f ∈ L1γθ, (4.2)

where Pi, i ≥ m, is as in (2.14) and ui, i ≥ m, is defined by

ui : [0, θ] → [0, θ]; ui(x) :=1

x+ iθ. (4.3)

(ii) Let µ be a probability measure on ([0, θ],B[0,θ]) such that µ is absolutelycontinuous with respect to the Lebesgue measure λθ and let h := dµ/dλθ

a.e. in [0, θ]. For any n ∈ N and A ∈ B[0,θ], we have

µ(

(Tθ)−n(A)

)

=

∫

A

Unf(x)dγθ(x), (4.4)

where f(x) := (log(1 + θ2))xθ+1θ

h(x), x ∈ [0, θ].

Proof. (i) Let Tθ,i denote the restriction of Tθ to the subinterval I(i) :=(

1(i+1)θ ,

1iθ

]

, i ≥ m, that is,

Tθ,i(x) =1

x− iθ, x ∈ I(i). (4.5)

Let C(A) := (Tθ)−1(A) and Ci(A) := (Tθ,i)

−1(A) for any A ∈ B[0,θ]. SinceC(A) =

⋃

iCi(A) and Ci ∩ Cj is a null set when i 6= j, we have

∫

C(A)f dγθ =

∑

i≥m

∫

Ci(A)f dγθ, f ∈ L1

γθ, A ∈ B[0,θ]. (4.6)

For any i ≥ m, by the change of variable x = (Tθ,i)−1(y) = ui(y), we

successively obtain

∫

Ci(A)f(x) dγθ(x) =

θ

log(1 + θ2)

∫

Ci(A)

f(x)

1 + xθdx

=1

log(1 + θ2)

∫

A

f (ui(y))

1 + ui(y)θ

θdy

(y + iθ)2

=

∫

A

Pi(y) f (ui(y)) dγθ(y). (4.7)

Now, (4.2) follows from (4.6) and (4.7).(ii) We will use mathematical induction. For n = 0, the equation (4.5) holdsby definitions of f and h. Assume that (4.5) holds for some n ∈ N. Then

µ((Tθ)−(n+1)(A)) = µ((Tθ)

−n((Tθ)−1(A)))

=

∫

C(A)Unf(x) dγθ(x). (4.8)

By the very definition of the Perron-Frobenius operator U we have∫

C(A)Unf(x) dγθ(x) =

∫

A

Un+1f(x) dγθ(x). (4.9)

Therefore,

µ(

(Tθ)−(n+1)(A)

)

=

∫

A

Un+1f(x)dγθ(x) (4.10)

which ends the proof.�

17

Let B([0, θ]) denote the collection of all bounded measurable functionsf : [0, θ] → C. A different interpretation is available for the operator Urestricted to B([0, θ]), which is a Banach space under the supremum norm.

Proposition 4.2. The operator U : B([0, θ]) → B([0, θ]) is the transitionoperator of both the Markov chain (sn,a)n∈N+

on ([0, θ],B[0,θ], γθ,a), for anya ∈ [0, θ], and the Markov chain (sl)l∈Z on ([0, θ]2,B2

[0,θ], γθ).

Proof. The transition operator of (sn,a)n∈N+takes f ∈ B([0, θ]) to the

function defined by

Ea (f(sn+1,a)| sn,a = s) =∑

i≥m

Pi(s)f(ui(s)) = Uf(s) for any s ∈ [0, θ],

(4.11)where Ea stands for the mean-value operator with respect to the probabilitymeasure γθ,a, whatever a ∈ [0, θ].

�

A similar reasoning is valid for the case of the Markov chain (sl)l∈Z.

Remark 4.3. In hypothesis of Proposition 4.1(ii) it follows that

µ(T−nθ (A)) − γθ(A) =

∫

A

(Unf(x)− 1)dγθ(x), (4.12)

for any n ∈ N and A ∈ B[0,θ], where f(x) := (log(1+θ2))xθ+1θ

h(x), x ∈ [0, θ].

The last equation shows that the asymptotic behavior of µ(T−nθ (A))− γθ(A)

as n → ∞ is given by the asymptotic behavior of the n-th power of thePerron-Frobenius U on L1

γθor on smaller Banach spaces.

5 Ergodicity of the associated RSCC

The facts presented in the previous sections lead us to a certain randomsystem with complete connections associated with the θ-expansion. To studythe ergodicity of this RSCC it becomes necessary to recall some definitionsand results from [7].

According to the general theory we have the following statement.

Definition 5.1. An homogeneous RSCC is a quadruple{(W,W), (X,X ), u, P} where

(i) (W,W) and (X,X ) are arbitrary measurable spaces;

(ii) u : W ×X → W is a (W ⊗X ,W)-measurable map;

18

(iii) P is a transition probability function from (W,W) to (X,X ).

For any n ∈ N+, consider the maps u(n) : W ×Xn → W , defined by

u(1) (w, x) := u(w, x),

u(n+1)(

w, x(n+1))

:= u(

u(n)(

w, x(n))

, xn+1

)

, n ≥ 1,

where x(n) = (x1, . . . , xn) ∈ Xn. We will simply write wx(n) for u(n)(w, x(n)).For every w ∈ W , r ∈ N+ and A ∈ X r, define

P1(w,A) := P (w,A),

Pr(w,A) :=

∫

X

P (w,dx1)

∫

X

P (wx1,dx2) . . .

∫

X

P (wx(r−1),dxr)χA(x(r)), r ≥ 2,

(5.1)where χA is the indicator function of the set A. Obviously, for n ∈ N+ fixed,Pr is a transition probability function from (W,W) to (Xr,X r).

By virtue of the existence theorem ([7], Theorem 1.1.2), for a givenRSCC {(W,W), (X,X ), u, P} there exists an associated Markov chain withthe transition operator U defined by

Uf(w) :=

∫

X

P (w,dx)f(wx), f ∈ B(W,W),

where B(W,W) is the Banach space of all bounded W-measurable complex-valued functions defined onW . Moreover, the transition probability functionof the associated Markov chain is

Q(w,B) :=

∫

X

P (w,dx)χB(wx) = P (w,Bw),

where Bw = {x ∈ X : wx ∈ B}, w ∈ W , B ∈ W. The iterates of theoperator U are given by

Unf(w) =

∫

Xn

Pn(w,dx(n))f(wx(n)), f ∈ B(W,W), n ∈ N+.

It follows that the n-step transition probability function is given by

Qn(w,B) = Pn(w,B(n)w ), w ∈ W, B ∈ W, n ∈ N+,

where B(n)w =

{

x(n) : wx(n) ∈ B}

. Hence the transition operator associatedwith the Markov chain with state space (W,W) and transition probabilityfunction Q is defined by

Uf(·) :=∫

W

Q(·,dw)f(w), f ∈ B(W,W). (5.2)

19

Its iterates are given by

Unf(·) =∫

W

Qn(·,dw)f(w), n ∈ N+, (5.3)

where Qn is the n-step transition probability function.Putting

Qn(w,B) =1

n

n∑

k=1

Qk(w,B)

for all n ∈ N+, w ∈ W and B ∈ W, it is clear that Qn is a transitionprobability function on (W,W). Let Un be the Markov operator associatedwith Qn.

Let (W,d) be a metric space and let L(W ) denote the Banach space ofall complex-valued Lipschitz continuous functions on W with the followingnorm:

‖f‖L := ‖f‖+ s(f), (5.4)

where

‖f‖ := supw∈W

|f(w)| , s(f) := supw′ 6=w′′

|f(w′)− f(w′′)|d(w′, w′′)

. (5.5)

Definition 5.2. (i) The operator U is said to be orderly with respect toL(W ) if and only if there exists a bounded linear operator U∞ on L(W )such that

limn→∞

‖Un − U∞‖L = 0.

(ii) The operator U is said to be aperiodic with respect to L(W ) if and onlyif there exists a bounded linear operator U∞ on L(W ) such that

limn→∞

‖Un − U∞‖L = 0.

(iii) The operator U is said to be ergodic with respect to L(W ) if and onlyif it is orderly and the range U∞(L(W )) is one-dimensional.

(iv) The operator U regular with respect to L(W ) if and only if it is ergodicand aperiodic.

Definition 5.3. The transition operator U of a Markov chain with statespace W is said to be a Doeblin-Fortet operator if and only if U takes L(W )into L(W ) boundedly with respect to ‖ · ‖L and there exist k ∈ N+, r ∈ [0, 1)and R < ∞ such that

s(Ukf) ≤ rs(f) +R‖f‖, f ∈ L(W ). (5.6)

Alternatively, the Markov chain itself is said to be a Doeblin-Fortet chain.

20

The definition below isolates a class of RSCCs, called RSCCs with con-traction, for which the associated Markov chains are Doeblin-Fortet chains.

Definition 5.4. An RSCC {(W,W) , (X,X ) , u, P} is said to be with con-traction if and only if (W,d) is a separable metric space, r1 < ∞, R1 < ∞and there exists j ∈ N+ such that rj < 1. Here

rj := supw′ 6=w′′

∫

Xj

Pj(w′,dx(j))

d(w′x(j), w′′x(j))d(w′, w′′)

(5.7)

andRj := sup

A∈X j

s(Pj(·, A)), j ∈ N+ (5.8)

where Pj is a transition probability function from (W,W) to(

Xj ,X j)

definedin (5.1).

We shall also need the following results.

Theorem 5.5. The Markov chain associated with an RSCC with contractionis a Doeblin-Fortet chain.

Lemma 5.6. Assume that the Markov operator U is aperiodic with respect toL(W ). Put T = U −U∞. Then we have U∞U∞ = U∞, TU∞ = U∞T = 0,Un = U∞ + T n, n ∈ N+. Moreover, there exist positive constants q < 1 andK such that

‖T n‖L ≤ Kqn, n ∈ N+. (5.9)

This immediately implies the validity of

‖Unf − U∞f‖L ≤ Kqn ‖f‖L , f ∈ L(W ), n ∈ N+. (5.10)

Definition 5.7. A Markov chain is said to be compact if and only if itsstate space is a compact metric space (W,d) and its transition operator is aDoeblin-Fortet operator.

Theorem 5.8. A compact Markov chain is orderly with respect to L(W )and there exists a transition probability function Q∞ on (W,W) such that

U∞f(·) =∫

W

Q∞(·,dw)f(w), f ∈ L(W ). (5.11)

Moreover, Q∞(·, B) ∈ E(1) for any B ∈ W and for any w ∈ W , Q∞(w, ·) isa stationary probability for the chain. (Here E(1) is the set of the eigenvaluesof modulus 1 of the operator U).

21

A criterion of regularity for a compact Markov chain is expressed inTheorem 5.9 in terms of the supports σn(w) of the transition probabilityfunctions Qn(w, ·), n ∈ N+.

Theorem 5.9. A compact Markov chain is regular with respect to L(W ) ifand only if there exists a point w∗ ∈ W such that

limn→∞

d (σn(w), w∗) = 0, w ∈ W. (5.12)

The application of this criterion is facilitated by the inter-relationshipamong the sets σn(w), n ∈ N+, which is given in the next lemma.

Lemma 5.10. For all m,n ∈ N and w ∈ W , we have

σm+n(w) =⋃

w′∈σm(w)

σn(w′), (5.13)

where the overline mean topological closure in W .

Now, we are able to study the following RSCC{(

[0, θ],B[0,θ]

)

, (Nm,P(Nm)) , u, P}

, (5.14)

where u : [0, θ] × Nm → [0, θ], u(s, i) = ui(s) is given in (4.3) and thefunction P (s, i) = Pi(s) given in (2.14) defines a transition probability from([0, θ],B[0,θ]) to (Nm,P ((Nm))). Here Nm = {m,m + 1, . . .}, m ∈ N+ andP (Nm) denotes the power set of Nm.

Whatever a ∈ [0, θ] the Markov chain (sn,a)n∈N associated with theRSCC (5.1) has the transition operator U , with the transition probabilityfunction

Q(s,B) =∑

{ i≥m|ui(s)∈B}Pi(s), s ∈ [0, θ], B ∈ B[0,θ]. (5.15)

Then Qn(·, ·) will denote the n-step transition probability function of thesame Markov chain.

Proposition 5.11. RSCC (5.14) is regular with respect to L([0, θ]). More-over there exist a stationary probability measure Q∞ = γθ and two positiveconstants q < 1 and K such that

∥

∥

∥

∥

Unf −∫ θ

0fdγθ

∥

∥

∥

∥

L

≤ Kqn ‖f‖L , n ∈ N+, f ∈ L([0, θ]), (5.16)

where

Unf(·) :=∫ θ

0Qn(·,ds)f(s) (5.17)

22

Proof. Since

d

dsui(s) =

−1

(s + iθ)2,

d

dsPi(s) =

iθ − 1/θ

(s + iθ)2− (i+ 1)θ − 1/θ

(s+ (i+ 1)θ)2,

for any s ∈ [0, θ] and i ≥ m it follows that

sups∈[0,θ]

∣

∣

∣

∣

d

dsui(s)

∣

∣

∣

∣

=1

(iθ)2

and

sups∈[0,θ]

∣

∣

∣

∣

d

dsPi(s)

∣

∣

∣

∣

< ∞.

Hence the requirements of Definition 5.4 of an RSCC with contraction aremet with j = 1. By Theorem 5.5 it follows that the Markov chain (sn,a)n∈Nassociated with this RSCC with contraction is a Doeblin-Fortet chain andits transition operator U is a Doeblin-Fortet operator. It remains to provethe regularity of U with respect to L([0, θ]). For this we have to prove theexistence of a point s∗ ∈ [0, θ] such that limn→∞ |σn(s)− s∗| = 0, for anys ∈ [0, θ], where σn(s) is the support of measure Qn(s, ·), n ∈ N+.

Let s ∈ [0, θ] be an arbitrarily fixed number and define

w1 := s, wn+1 :=1

wn +mθ, n ∈ N+. (5.18)

We have wn ∈ [0, θ] and letting n → ∞ in (5.18) we get

wn → s∗ :=−1 +

√1 + 4θ2

2θ.

Clearly, wn+1 ∈ σ1(wn) and Lemma 5.10 and an induction argument showthat wn ∈ σn(s), n ∈ N+. Thus

d (σn(s), s∗) ≤ |wn − s∗| → 0, n → ∞

where d stands for the Euclidian distance on the line. Now, the regularityof U with respect to L([0, θ]) follows from Theorem 5.9.

From (5.10) and Theorem 5.8 there exist a stationary probability mea-sure Q∞ and two constants q < 1 and K such that

‖Unf − U∞f‖L ≤ Kqn ‖f‖L , n ∈ N+, f ∈ L([0, θ]) (5.19)

23

where Unf is as in (5.17) and

U∞f =

∫ θ

0f(x)Q∞(dx). (5.20)

Here Q∞ = γθ is the invariant probability measure of the transformationTθ in (1.11), i.e., Q∞ has the density ρθ = 1/(x +mθ), x ∈ [0, θ], with thenormalizing factor 1/ log(1 + θ2).

�

Remark 5.12. Another way to put this is that ρθ is the eigenfunction ofeigenvalue 1 of the Perron-Frobenius operator U .

6 A Gauss-Kuzmin-type theorem

Now, we may determine the limit of the sequence (µ(T nθ < x))n∈N+

asn → ∞ and give the rate of this convergence.

Theorem 6.1. (A Gauss-Kuzmin-type theorem for Tθ) Let ([0, θ],B[0,θ], Tθ)be as in Definition 1.1(i).

(i) For a probability measure µ on ([0, θ],B[0,θ]), let the assumption (A)as follows:

(A) µ is non-atomic and has a Riemann-integrable density.

Then for any probability measure µ which satisfies (A), the followingholds:

limn→∞

µ(T nθ < x) =

1

log(1 + θ2)log((mθ + x)θ), x ∈ [0, θ]. (6.1)

(ii) In addition to assumption of µ in (i), if the density of [0, θ] ∋ x 7→µ([0, x]) is Lipschitz continuous, then there exist two positive constantsq < 1 and K such that for any x ∈ [0, θ] and n ∈ N+, the followingholds:

limn→∞

µ(T nθ < x) =

1 + αqn

log(1 + θ2)log((mθ + x)θ), (6.2)

where α := α(µ, n, x) with |α| ≤ K.

24

As a consequence, the n-th error term en(θ, µ;x) of the Gauss-Kuzminproblem is obtained as follows:

en(θ, µ;x) =αqn

log(1 + θ2)log((mθ + x)θ). (6.3)

Proof.Let Tθ be as in (1.11). By Proposition 4.1(ii), we have

µ(

(Tθ)−n(A)

)

=

∫

A

Unf0(x)ρθ(x)dx for any n ∈ N, A ∈ B[0,θ] (6.4)

where f0(x) = xθ+1θ

(dµ/dλθ)(x) for x ∈ [0, θ]. If dµ/dλθ ∈ L([0, θ]), by(5.20) we have

U∞f0 =

∫ θ

0f0(x)Q

∞(dx) =

∫ θ

0f0(x) γθ(dx) =

1

log(1 + θ2). (6.5)

Taking into account (5.19), there exist two constants q < 1 and K suchthat

‖Unf0 − U∞f0‖L ≤ Kqn ‖f0‖L , n ∈ N+. (6.6)

Furthermore, consider the Banach space C([0, θ]) of all real-valued continu-ous functions on [0, θ] with the norm ‖f‖ := supx∈[0,θ] |f(x)|. Since L([0, θ])is a dense subspace of C([0, θ]) we have

limn→∞

‖(Un − U∞)f‖ = 0 for all f ∈ C([0, θ]). (6.7)

Therefore, (6.7) is valid for a measurable function f0 which is Q∞-almostsurely continuous, that is, for a Riemann-integrable function f0. Thus, wehave

limn→∞

µ (T nθ < x) = lim

n→∞

∫ x

0Unf0(u)ρθ(u) du (6.8)

=1

log(1 + θ2)

∫ x

0ρθ(u) du (6.9)

=1

log(1 + θ2)log((mθ + x)θ). (6.10)

Hence (6.1) is proved. �

25

Remark 6.2. Since the Lebesgue measure λ satisfies assumptions in both(i) and (ii) of Theorem 6.1, (6.1) and (6.2) hold for the case µ = λ. HenceTheorem 6.1 gives the solution of the Gauss-Kuzmin problem for the pair(Tθ, µ) instead of (τ, λ) in (1.3).

Remark 6.3. Until now, the estimate of the convergence rate remains anopen question. To obtain a better estimate of the convergence rate involved,we may use a Wirsing-type approach as in [19, 20, 21].

References

[1] Bhattacharya, R. and Goswami, A., A class of random continued frac-tions with singular equillibria, Perspectives in Statistical Sciences, edsA.K.Basu et al (2000), Oxford University Press.

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